Having bought "The First English Dictionary (1604) when it was reprinted in 2007, I feel I've been had. I've looked at a third of this amazing 1580 book, and I can confirm it's extremely exciting. The annotations are consistent with the book's editor preparing a new edition, adding many new words (though without definitions).
I can also confirm that the annotations are not in the hand of William Shakspere of Stratford. Truth be told, we have nothing in his handwriting to compare this with. Contrary to what you may have thought, even the six signatures we supposedly have are so different from one another than they weren't made by the same person.
Other than circular reasoning, there is absolutely no evidence that he was literate. He was an unusually successful (if unscrupulous) businessman. He was fined for hoarding grain during a famine (the original version of his Stratford momument shows him holding-- or hoarding-- a sack of grain; only more than a century later were the quill and paper added). No, we have no evidence he attended the Stratford grammar school.
The whole misunderstanding is based on our ignorance about the importance of authorial anonymity in Elizabethan England.